Detailed calculation of lepton flavor violating muon-electron conversion rate for various nuclei

The coherent $\mu -e$ conversion rates in various nuclei are calculated for general lepton flavor violating interactions. We solve the Dirac equations numerically for the initial state muon and the final state electron in the Coulomb force, and perform the overlap integrations between the wave functions and the nucleon densities. The results indicate that the conversion branching ratio increases for a light nucleus up to the atomic number $Z\sim 30,$ is largest for $Z=30\mathrm{}–60,$ and becomes smaller for a heavy nucleus with $Z\gtrsim 60.$ We also discuss the uncertainty from the input proton and neutron densities. The atomic number dependence of the conversion ratio calculated here is useful to distinguish theoretical models with lepton flavor violation.